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Complex Numbers. Chapter 12. The Imaginary Number j. Previously, when we encountered an equation like x 2 + 4 = 0, we said that there was no solution since solving for x yielded . There is no real number that can be squared to produce -4.

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complex numbers
Complex Numbers

Chapter 12

slide2

The Imaginary Number j

Previously, when we encountered an equation like x2 + 4 = 0, we said that there was no solution since solving for x yielded

There is no real number that can be squared to produce -4.

Ah… but mathematicians were not satisfied with these so-called unsolvable equations. If the set of real numbers was not up to the task, they would define an expanded system of numbers that could handle the job!

Hence, the development of the set of complex numbers.

slide3

Definition of a Complex Number

The imaginary number j is defined as , where j2= .

A complex number is a number in the form x+ yj, where x and y are real numbers. (x is the real part and yj is the imaginary part)

slide5

The rectangular Form of a Complex Number

Each complex number can be written in the rectangular form x + yj.

Example

Write the complex numbers in rectangular form.

slide6

Addition & Subtraction of Complex Numbers

To add or subtract two complex numbers, add/subtract the real parts and the imaginary parts separately.

Example #1

slide7

Addition & Subtraction of Complex Numbers

Example #2

Simplify and write the result in rectangular form.

slide8

Multiplying Complex Numbers

Multiply complex numbers as you would real numbers, using the

distributive property or the FOIL method, as appropriate. Simplify your

answer, keeping in mind that j2= -1.

Always write your final answer in rectangular form, x + yj.

Example #1

slide9

MultiplyingComplex Numbers (continued)

Example #2

Simplify and write the result in rectangular form.

slide10

MultiplyingComplex Numbers (continued)

Example #3

Simplify and write the result in rectangular form.

slide11

Be careful

Multiplying with Complex Numbers (continued)

Example #4

Simplify and write the result in rectangular form.

slide12

Multiplying with Complex Numbers (continued)

Example #5

Simplify each expression.

slide13

Multiplying with Complex Numbers (continued)

Example #6

Simplify and write the result in rectangular form.

slide14

Powers of j

Complete the following:

“What pattern do you observe?”

slide15

Powers of j

Examples

Simplify and write the result in rectangular form.

Note: If a complex expression is in simplest form, then the only power of jthat should appearin the expression is j1.

slide16

Dividing Complex Numbers

  • For a quotient of complex numbers to be in rectangular form, it cannot have jin the denominator.
  • Scenario 1: The denominator of an expression is in the form yj
  • Multiply numerator and denominator by j
  • Then use the fact that j2 = -1 to simplify the expression and write in rectangular form.
slide18

Dividing Complex Numbers (continued)

Example #2

Write the quotient in rectangular form.

slide19

Complex Conjugates

Pairs of complex numbers in the form x + yj and x – yj are called complex conjugates.

These are important because when you multiply the conjugates

together (FOIL), the imaginary terms drop out, leaving only x2 + y2.

We will use this idea to simplify a quotient of complex numbers in rectangular form.

slide20

Complex Conjugates (continued)

  • Scenario 2: The denominator of an expression is in the form x+yj
  • Multiply numerator and denominator by the conjugate of the denominator
  • Then use the fact that j2 = -1 to simplify the expression and write in rectangular form.
slide22

Complex Conjugates (continued)

Example #2

Write the quotient in rectangular form.

slide23

Graphical Representation of Complex Numbers

A complex number can be represented graphically as a point in the rectangular coordinate system.

For a complex number in the form x + yj, the real part, x, is the x-value and the imaginary part, y, is the y-value.

In the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis.

slide24

Graphical Representation of Complex Numbers

Graph the points in the complex plane:

A: -3 + 4j

B: -j

C: 6

D: 2 – 7j

slide25

Earlier, we saw that a point in the plane could be located by polar coordinates, as well as by rectangular coordinates, and we learned to convert between polar and rectangular.

Polar Coordinates

slide26

imaginary

r

real

Now, we will use a similar technique with complex numbers, converting between rectangular and polar form*.

*The polar form is sometimes called the trigonometric form.

We’ll start by plotting the complex number x + yj, drawing a vector from the origin to the point.

Polar Form of a Complex Number

To convert to polar form, we need to know:

slide27
The polar form is found by substituting the values of x and y into the rectangular form.

or

A commonly used shortcut notation for the polar form is

slide29
Example

Represent the complex numbers graphically and give the polar form of each.

1) 2 + 3j

2) 4

slide30
Example

Represent the complex numbers graphically and give the polar form of each.

3)

4)

slide31
Example

The current in a certain microprocessor circuit is given by

Write this in rectangular form.

slide32

The exponential form of a complex number is written as

This form is used commonly in electronics and physics applications, and is convenient for multiplying complex numbers (you simply use the laws of exponents).

Remember, from the chapter on exponential and logarithmic equations, that e is an irrational number that is approximately equal to 2.71828. (It is called the natural base.)

The Exponential Form of a Complex Number

slide33
The Exponential Form of a Complex Number

 in radians

known as Euler’s Formula

slide34
Example

Write the complex number in exponential form.

slide35
Example

Write the complex number in exponential form.

slide36
Example

Write the complex number in exponential form.

slide37
Example

Express the complex number in rectangular and polar forms.

slide38

Rectangular:

Polar:

Exponential:

We have have now used three forms of a complex number:

So we have,